We show the calculated band structure and DOS in Fig. REF . The electronic DOS near the Fermi level are dominated by Rh-\(4d\)  and Sb-\(5p\)  orbitals. The La-5\(d\)  orbitals have only negligible contribution. The La-4\(f\)  states are approximately 2 eV above the Fermi level and are irrelevant. Since the crystal structure contains inversion center, the spin degeneracy is preserved with SOC considered. As the crystal structure of the compound contains 4 formula units of La\(_2\) Rh\(_3\) Sb\(_4\) , its band structure is quite complicated too. There are as many as 12 doubly degenerate bands crossing the Fermi level, thus giving rise to 12 doubly degenerate Fermi surface sheets. Around \(\Gamma \) , semi-flat bands appear along \(\Gamma \) -Z and \(\Gamma \) -X. As a result, a small DOS peak appear at the Fermi level in the pristine compound. These semi-flat bands may be related to reduced dimensionality in certain directions, as two of these bands do not cross the Fermi level along U-R and Z-T. Therefore, the DOS peak at \(E_F\)  in the pristine La\(_2\) Rh\(_3\) Sb\(_4\)  may be interpreted as van-Hove singularity like, and yields a large \(N(E_F)=7.31\)  states/(f.u.\(\cdot \) eV), or equivalently \(\gamma _{\mathrm {calc}}=\)  17.24 mJ/mol\(\cdot \) K\(^2\) . Compared to experimental value assuming \(\gamma _n=\gamma _{\mathrm {calc}}(1+\lambda _{\mathrm {elph}})\) , we obtain a small electron-phonon coupling constant \(\lambda _{\mathrm {elph}}=\)  0.12, which cannot account for the observed \(T_c\) . Alternatively, if we take the inter-site Rh4 impurities as charge dopants, we can estimate the amount of electron doping to be about 1 electron/unit cell[1]}. Employing the rigid band approximation, the DOS at the Fermi level is found to be \(N(E_F)=\)  6.0 states/(f.u.\(\cdot \) eV), or equivalently \(\gamma _{\mathrm {calc}}=\)  14.14 mJ/mol\(\cdot \) K\(^2\)  for the doped system. Thus, the electron-phonon coupling constant of the La\(_2\) Rh\(_{3+\delta }\) Sb\(_4\)  can be estimated to be \(\lambda _{\mathrm {elph}}\)  = 0.375. \(T_c\)  can be estimated according to the McMillan equation in the weak electron-phonon coupling regime[2]},
\(T_c=\frac{\Theta _D}{1.45}\exp [-\frac{1.04(1+\lambda _{\mathrm {elph}})}{\lambda _{\mathrm {elph}}-\mu ^*(1+0.62\lambda _{\mathrm {elph}})}],\) 
